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About a decade ago, supported by a NSF- CCD grant, an undergraduate-level nonlinear dynamics course was developed at Cal Poly. The novel features of the course were the laboratory component and its interdisciplinary nature. A unique feature of the lab component is that the experiments utilize readily available equipment found in a typical undergraduate physics program. The inspiration for the course arose from the fact that the traditional undergraduate science and engineering curriculum emphasizes analytical solutions of differential equations which are not very useful in most real systems encountered by students later in life. The elegant geometrical methods and visualization techniques of nonlinear dynamics have not yet been incorporated into the undergraduate curriculum.
The nonlinear dynamics course is offered as an upper division elective course to all science and engineering majors. So far it has been offered 6 times and taken by around 100 students, 55% of which were physics majors and the rest mostly from a variety of engineering departments. The course has a three 1 hour lecture component and a three hour lab component offered over a 10 week quarter. The lectures emphasize geometrical methods and visualization tools such as phase space, fixed points, bifurcations, limit cycles and attractors. The textbook used for the course is “Nonlinear Dynamics and Chaos” by S. Strogatz which is at the appropriate level and has an interdisciplinary approach. The experiments that were developed at Cal Poly follow the lecture material closely and teach data acquisition, data display, and analysis techniques such as power spectra, Poincare sections and return maps on a variety of systems from different fields. The experiments can be downloaded at the web site www.calpoly.edu/~nsungar/nonlinear.html and more information on the course can be found in AJP 69 (5), 591-597 (2001). A major feature of the lab component is that after doing prescribed experiments for seven weeks, students are required to complete a three week project on a system of their choice. Initially, the goal of the project component was to allow students apply their knowledge on a system in their field. Over the many offerings of the course, it was also realized that the project component provides a unique experience on an open-ended problem and the students show great enthusiasm and effort. Two weeks before they start the project, a collection of literature on experiments (including computational experiments) on nonlinear systems are made available to the students. They are also encouraged to talk to professors in their own departments and do a literature search. Another possibility that is presented to the students is to expand on and do a more sophisticated analysis of one of the prescribed experiments done earlier in the quarter. Of all the students who took the course, only 22% chose to expand one of the prescribed experiments. It was also noted that 38% of the projects were computational and the rest experimental. The projects involve construction of new equipment, assembling of equipment and programming and can be open ended. Students have to choose, plan, perform and report with minimum supervision from faculty. It must be recognized however that the supervision of the projects is faculty-intensive. All faculty involved in the course (five of us in the Physics Department), even those who are not teaching that quarter participate in supervising projects. In addition, several technicians in the physics department assist students in finding or ordering equipment for their projects. The students are expected to write a formal report on their project. About 50% of the students actually bring their project to a successful completion. However, almost all students show great effort and creativity and learn to deal with problems that arise in designing and conducting an experiment. The textbook which has examples from many disciplines also provides ideas for projects. For example, one of the students has built the chaotic waterwheel described in the textbook while others attempted to build the bead on a tilted wire example. We have also noticed that there are several “popular” projects, taken on by multiple students. These include the double pendulum, the chaotic bouncing ball, the buckling beam and construction of chaotic electronic circuits.. In a few cases, physics majors, after taking the course have done research with faculty on projects related to nonlinear dynamics.
In summary, we believe that such a course incorporating the geometrical methods for dynamical systems represented by differential equations provide valuable addition to student’s education. The project component highly motivates the students and allows them to work on an open ended problem and apply the techniques they learn in class. Although the project component is intensive in faculty and technician time, the equipment required to build such a course is readily available in most Physics departments and, if not, could be acquired at low cost. A possible improvement would be to allow longer time for the projects which could easily be done at institutions following a semester system.
Nilgun Sungar is a professor in the Physics Department at California Polytechnic State University, San Luis Obispo. Her current research interests are in biophysics and nonlinear dynamics.